I applaud your asking permission to recover the wood, so many brag openly about "liberating", "snagging" or just plain stealing bow wood without a second thought.
Your wood is not bow quality wood. There is a possibility you can cut the longer, narrow limb at the fork, split or cut it in half length wise and have two billet pieces you could splice together in the handle to make a bow.
This limb piece will be full of knots with very thick sapwood and very little heart wood, a challenge for sure but doable.
I would split it in half, seal the ends really well, leave the bark on, spray the bark with a strong bug spray to keep the wood wasps at bay and let it sit for a year or so.
If you make a bow from it you will need a heat gun to straighten it which isn't that hard to do.
I suspect you will make a bow with a combination of heartwood and sapwood, maybe only heart wood if the sapwood is really thin.
In my pile your piece would be used for turning tool handles on my lathe or for firewood but it doesn't hurt to give it a try as bow wood.